On the least singular value of random symmetric matrices

TL;DR

This paper establishes probabilistic bounds on the smallest singular value of randomly perturbed symmetric matrices with bounded entries, under general distributional assumptions, using inverse concentration results.

Contribution

It provides a general probabilistic bound on the least singular value of symmetric matrices with random perturbations, extending previous results to broader settings.

Findings

Probability that the smallest singular value is very small is polynomially bounded

The bounds hold under very general assumptions on the distribution of entries

Uses an inverse concentration result for quadratic forms

Abstract

Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal entries $x_{ij}$ are iid copies of a random variable $\xi$. Under a very general assumption on $\xi$, we show that for any $B>0$ there exists $A>0$ such that $P(\sigma_n(M_n)\le n^{-A})\le n^{-B}$. The proof uses an inverse-type result concerning concentration of quadratic forms, which is of interest of its own.